Hidden Markov Models: Fundamentals and Applications Part 1: Markov Chains and Mixture Models


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1 Also appears n the Onlne Symposum for Electroncs Engneer Hdden Marov Models: Fundamentals and Applcatons Part : Marov Chans and Mxture Models Valery A. Petrushn Center for Strategc Technology Research Accenture 3773 Wllo Rd. Northbroo, Illnos 6006, USA. Abstract The obectve of ths tutoral s to ntroduce basc concepts of a Hdden Marov Model (HMM) as a fuson of more smple models such as a Marov chan and a Gaussan mxture model. The tutoral s ntended for the practcng engneer, bologst, lngust or programmer ho ould le to learn more about the above mentoned fascnatng mathematcal models and nclude them nto one s repertore. Ths lecture presents Marov Chans and Gaussan mxture models, hch consttute the prelmnary noledge for understandng Hdden Marov Models. Introducton Why t s so mportant to learn about these models? Frst, the models have proved to be ndspensable for a de range of applcatons n such areas as sgnal processng, bonformatcs, mage processng, lngustcs, and others, hch deal th sequences or mxtures of components. Second, the ey algorthm used for estmatng the models the socalled Expectaton Maxmzaton Algorthm  has much broader applcaton potental and deserves to be non to every practcng engneer or scentst. And last, but not least, the beauty of the models and algorthms maes t orthhle to devote some tme and efforts to learnng and enoyng them. Introducton nto Marov chans. Pavng the path to the shrne problem Welcome to the planet Medan! Medan s the captal planet of the Stochastc Empre, hch conquered all orlds of the observable part of the Unverse and spread ts nfluence to the hdden parts. Ruled by the Glorous and Far ruler, Emperor Probabl the Great, the Empre reached the most flourshng state of commerce, scence and ndustry ever. Welcome to BayesCty, the captal of the Empre! Today, the Emperor s heralds had the honor of announcng a ne gant proect. The four maor Provnces of the Empre have to pave the paths from ther resdence ctes on Medan to the St. ely Hood Shrne, located on the Mean Square n Fgure.. Pavng problem BayesCty. Each path s,000 mles long and has to be paved usng large tles of the follong precous stones: ruby (red), emerald (green) and topaz (blue) n accordance to ts
2 Provnce s random process hch states that the color of next tle depends only on the color of the current tle. The square around the St. ely Hood Shrne ll be secretly dvded nto sectors and paved by the teams from all four Provnces over one nght (Fgure.). A plgrm ll be alloed to enter the Shrne to enoy the ve of the Bayesan Belef Netor, the log that served as a char to ely Hood, and the other relcs, and also to tal to the hgh prests of scence durng the hgh confdence tme ntervals f and only f he or she can determne for three sectors of pavement around the Shrne hch sector has been paved by the team of hch Provnce. Imagne you are a plgrm ho ourneyed more than,000 mles on foot to see the Stochastc Empre s relcs. Ho can you solve the problem?. Marov chans We deal th a stochastc or random process hch s characterzed by the rule that only the current state of the process can nfluence the choce of the next state. It means the process has no memory of ts prevous states. Such a process s called a Marov process after the name of a promnent Russan mathematcan Andrey Marov (8569). If e assume that the process has only a fnte or countable set of states, then t s called a Marov chan. Marov chans can be consdered both n dscrete and contnuous tme, but e shall lmt our tutoral to the dscrete tme fnte Marov chans. (a) (b) π = ( ) π = ( ) Fgure.. Marov chan models for a based con (a), and the pavng model M N Such chans can be descrbed by dagrams (Fgure.). The nodes of the dagram represent the states (n our case, a state corresponds to a choce of a tle of a partcular color) and the edges represent transtons beteen the states. A transton probablty s assgned to each edge. The probabltes of all edges outgong from a node must sum to one. Besde that, there s an ntal state probablty dstrbuton to defne the frst state of the chan. M = π, A π = ( π ), π,..., π N { a }, =, N + = + (.) Thus, a Marov chan s unquely defned by a par (.), here π s the vector of ntal probablty dstrbuton and A s a stochastc transton matrx. The Marov process characterstc property s represented by (.). Fgure. presents Marov chan models for a based con and tle generaton..3 Tranng algorthm ( q q,..., q ) P( q q ) P (.) We assume that each Provnce uses ts on Marov chan to generate sequences of tles. We also assume that all tles have ther ordnal numbers carved on them and e have no problem determnng the order of tles n the sequence (n spte of the fact that the tles cover the todmensonal surface of the path). The resultng paths could be vsually dfferent for pavements of dfferent Provnces. For example, North Provnce, hch s famous for ts deposts of emerald, mght have larger probabltes for transtons to emerald (green) state and generate greensh paths. But the dfferences among models could be rather small and the paths produced by the models could loo very smlar, especally for the paths of short length. That s hy e have to fnd a formal approach for estmatng the model's parameters based on
3 data. Thans to the Emperor, e have enough data to tran our models ,000 mles of data per model! If a tle has the dameter of one yard and the tles are lned up n ros of 0 tles per ro then e have 35,00 tles per mle. To estmate parameters of a model th 3 states, e need to calculate values: a 3element vector of ntal state probabltes and a 3 by 3 state transton probablty matrx. et Q = qq... q be a tranng sequence of states. As estmatons for the ntal state probabltes, e use frequences of tles c n the tranng sequence (.3). To estmate transton probabltes, e have to count the number of pars of tles for each combnaton of colors c (,=,N) and then normalze values of each ro (.4). Table. presents three estmates of parameters for the ncreasng length of the tranng sequence. Table.. Marov chan tranng results π = True =000 =0000 =3500 ( ) c, π = a c c = N = N, =, = N N N = = c = (.3) (.4) π = ( ) π = ( ) π = ( ) No you no hat to do! You should al along each path, collect data and use (.3) and (.4) to create Marov chans for each Provnce..4 Recognton algorthm et us assume that you have already bult the models for each Provnce: MN, ME, MS, and MW. Ho can you use them to determne hch Provnce s team paved a sector Q? A natural approach to solve ths problem s to calculate the condtonal probablty of the model M for the gven sequence of tles Q: P(M Q). Usng the Bayes Rule (.5), e can see that P(Q) does not depend on the model. Thus, e should pc up the model, hch maxmzes the numerator of the formula (.5). P( M P( Q M ) P( M ) Q) = P( Q) here P(M Q) s the posteror probablty of model M for the gven sequence Q; P(Q M ) s the lelhood of Q to be generated by M ; P(M ) s the pror probablty of M ; P(Q) s the probablty of Q to be generated by all models. Ths crteron s called maxmum aposteror probablty (MAP) estmaton and can be formally expressed as (.6). Hoever, t s often the case that the pror probabltes are not non or are unform over all the models. Then e can reduce (.6) to (.7). Ths crteron s called maxmum lelhood (M) estmaton. M M = arg max{ P( Q M * M = argmax{ P( Q M )} ) P( M * M )} (.5) (.6) (.7)
4 Ho can you calculate the lelhood? et us consder the sequence 33 and trace on the dagram n Fgure.b to see ho ths sequence could be generated. The frst state can be generated th the probablty π 3, the transton probablty from 3 to s a 3, and from to 3 s a 3. The probablty of the sequence s equal to the product of these three factors (.8). P Q M ) = π a a = (.8) ( = Generalzng our dervaton for the sequence of arbtrary length e obtan (.9). Typcally the logarthm of lelhood or loglelhood s used for calculatons (.0). P( Q M ) = π a a... a = π q qq qq3 q q q log P( Q M ) = logπ q + loga = Table. presents the accuracy of recognton of,000 sequences of dfferent length. The test sequences ere generated by the true models and recognzed by the estmated models. For recognton e used the M crteron,.e. e calculated the loglelhood of the gven sequence for each of four models and pced up the model th the maxmal value. The set of models as traned on onemle data for each model. You can see that the accuracy s poor for the short sequences and gradually mproves th grong length of sequences. For relable recognton e need sequences of more than 00 symbols. q q = a q q (.9) (.0) Table.. Recognton accuracy for test sequences of dfferent length ength Accuracy (%) True models: North East South West π = ( ) π = ( ) = ( ) π = π ( ) Estmated onemle models: North East South West π = ( ) π = ( ) π = ( ) π = ( 3 3 4) Generalzatons To mportant generalzatons of the Marov chan model descrbed above are orth to mentonng. They are hghorder Marov chans and contnuoustme Marov chans. In the case of a hghorder Marov chan of order n, here n >, e assume that the choce of the next state depends on n prevous states, ncludng the current state (.). P ( q ) ( ) + n q, q,..., q,..., q+ n = P q+ n q,..., q + n (.) In the case of a contnuoustme Marov chan, the process ats for a random tme at the current state before t umps to the next state. The atng tme at each state can have ts on probablty densty functon or the common functon for all states, for example, an exponental dstrbuton (.). f T λt ( t) = e t 0 λ (.)
5 .6 References and Applcatons The theory of Marov chans s ell establshed and has vast lterature. Tang all responsblty for my slant choce, I ould le to menton only to boos. Norrs boo [] s a concse ntroducton nto the theory of Marov chans, and Steart s boo [] pays attenton to numercal aspects of Marov chans and ther applcatons. Both boos have some recommendatons and references for further readng. Marov Chans have a de range of applcatons. One of the most mportant and elaborated areas of applcatons s analyss and desgn of queues and queung netors. It covers a range of problems from productvty analyss of a carash staton to the desgn of optmal computer and telecommuncaton netors. Steart s boo [] gves a lot of detals on ths type of modelng and descrbes a softare pacage that can be used for queung netors desgn and analyss ( Hstorcally, the frst applcaton of Marov Chans as made by Andrey Marov hmself n the area of language modelng. Marov as fond of poetry and he appled hs model to studes of poetc style. Today you can have fun usng the Shannonzer hch s a Webbased program (see that can rerte your message n the style of Mar Tan or es Carroll. Another example of Marov chans applcaton n lngustcs s stochastc language modelng. It as shon that alphabet level Marov chans of order 5 and 6 can be used for recognton texts n dfferent languages. The ord level Marov chans of order and are used for language modelng for speech recognton ( DNA sequences can be consdered as texts n the alphabet of four letters that represent the nucleotdes. The dfference n stochastc propertes of Marov chan models for codng and noncodng regons of DNA can be used for gene fndng. GeneMar s the frst system that mplemented ths approach (see The other group of models non as branchng processes as desgned to cover such applcaton as modelng chan reacton n chemstry and nuclear physcs, populaton genetcs ( and even game analyss for such games as baseball ( curlng, etc. And the last applcaton area, hch I d le to menton, s so called Marov chan Monte Carlo algorthms. They use Marov chans to generate random numbers that belong exactly to the desred dstrbuton or, speang n other ords, they create a perfectly random samplng. See the for addtonal nformaton about these algorthms. Introducton nto Gaussan mxture models. Revealng the old mon s secret problem All respectful famles on Medan have ther tradtonal vast gardens of sculptures. The gardens, hch are covered by sand composed of mcroscopc grans of precous stones, exhbt both the sculptures of concrete thngs and persons and abstract concepts. The color of the sand s as mportant as the artstc and scentfc value of sculptures. The sand recpes are ept n secret and passed from one generaton to another. A ne rch man (let us call hm Mr. Nerch), ho nterpreted the slogan eep off Medan! as permsson to explore and explot poor developed suburbs of the Empre, returned to the captal planet and settled hmself n a prestgous area. But hen he started to desgn hs garden of sculptures, he ran nto a problem of sand recpe. After several
6 unsuccessful attempts to buy a recpe he decded to steal some sand from the orshop of an old mon ho served as a sandeeper for a noble and respectful, but not too rch famly, and reveal the secret of the recpe usng the stolen sand as data. Mr. Nerch s helpers measured the avelength of reflected lght for 0,000 sand grans. Fgure. presents the hstogram for the data. Three concentratons of data near 400 nm, 40 nm and 450 nm allo us to form a hypothess that sand s a mxture of three components. Fgure.. Hstogram of sand data. et us pretend that you are the chef scentst at Allempre Consultng, Inc., and Mr. Nerch pays you a fortune to decode the data. Ho can you determne the characterstcs of each component and proporton of the mxture? It seems that stochastc modelng approach called Gaussan Mxture Models can do the ob.. Gaussan mxture model et us assume that components of the mxture have Gaussan dstrbuton th mean µ and varance σ and probablty densty functon (.). The choce of the Gaussan dstrbuton s natural and very common hen e deal th a natural obect or phenomenon, such as sand. But the mxture model approach can be appled to the other dstrbutons from the exponental famly. σ π ( x µ ) ( ;, ) σ f x µ σ = e (.) We let o,o,,o denote our observatons or data ponts. Then each data pont s assumed to be an ndependent realzaton of the random varable O th the threecomponent mxture probablty densty functon (.) and loglelhood functon (.3). Wth the maxmum lelhood approach to the estmaton of our model M = ν, N ( µ, σ ),..., N ( µ, σ ), e need to fnd such values of ν, µ and σ that maxmze the functon (.3). To solve the problem, the Expectaton Maxmzaton (EM) Algorthm s used. f ( o; M ) = ν f = log ( M ) = ( o; µ, σ ) log( = = The EM Algorthm s also non as the Fuzzy mean algorthm. The ey dea of the algorthm s to assgn to each data pont o a vector that has as many elements as there are components n the mxture (three n our case). Each element of the vector (e shall call them eghts) represents the confdence or probablty that the th data pont o belongs to the ν f ( o ν = ; µ, σ )) = (.) (.3)
7 th mxture component. All eghts for a gven data pont must sum to one (.4). For example, a tocomponent mxture eght vector (, 0) means that the data pont certanly belongs to the frst mxture component, but (, 0.7) means than the data pont belongs more to the second mxture component than to the frst one. = P( N( µ, σ ) o ), = (.4) For each teraton, e use the current parameters of the model to calculate eghts for all data pont (expectaton step or Estep), and then e use these updated eghts to recalculate parameters of the model (maxmzaton step or Mstep). et us see ho the EM algorthm ors n some detals. Frst, e have to provde an ntal approxmaton of model M (0) usng one of the follong ays: () Parttonng data ponts arbtrarly among the mxture components and then calculatng the parameters of the model. () Settng up the parameters randomly or based on our noledge about the problem. After ths, e start to terate E and Msteps untl a stoppng crteron gets true. For the Estep the expected component membershp eghts are calculated for each data pont based on parameters of the current model. Ths s done usng the Bayes Rule that calculates the eght as the posteror probablty of membershp of th data pont n th mxture component (.5). Compare the formula (.5) to the Bayes Rule (.5). h= For the Mstep, the parameters of the model are recalculated usng formulas (.6), (.7) and (.8), based on our refned noledge about membershp of data ponts. A nce feature of the EM algorthm s that the lelhood functon (.3) can never decrease; hence t eventually converges to a local maxmum. In practce, the algorthm stops hen the dfference beteen to consecutve values of lelhood functon s less than a threshold (.9). ( t+ ) ( t ) log ( M ) log ( M ) < ε (.9) In order to better understand ho the EM algorthm ors, let us trace t for the case of to mxture components th a small amount of data ponts. Suppose e have 0 data ponts that are arbtrarly parttoned beteen to components (see Fgure.3a). The ntal model has large varances and means that le close to each other (see Table.). Usng ths model, the algorthm calculates a ne model th larger dstance beteen means and smaller varances. After teratons the algorthm produces the model that maxmzes the lelhood functon and optmzes the membershp probablty dstrbutons for each data pont (see Fgure.3 and Table.). The resultng soluton s depcted as to green bellshaped curves on Fgure.3d. You can see that e have perfect mxture proporton, and good approxmaton for the mean of the frst component and varance of the second component. In general, the results are amazngly good for a data set of ten ponts. = = ν f ( o ; µ, σ ) ν f ( o ; µ, σ ) ( t+ ) ν = =, µ σ ( t+ ) ( t+ ) = = = = = o = ( o µ ) h = h h (.5) (.6) (.7) (.8)
8 Fgure.3. EM algorthm convergence. Table. EM algorthm convergence (0 ponts) Applyng the EM algorthm to Mr. Nerch s data, e obtan the soluton presented n Table.. Due to a rather good ntal model t too only 7 teratons to converge. The old mon s secret s revealed! Table.. Mon s secret problem
9 The mxture model approach and the EM algorthm have been generalzed to accommodate multvarate data and other nd of probablty dstrbutons. Another method of generalzaton s to splt multvarate data nto several sets or streams and create a mxture model for each stream, but have the ont lelhood functon. The specal beauty of mxture models s that they or n cases hen the components are heavly ntersected. For example, for todmensonal data of 000 ponts and threecomponent mxture th to components havng equal means but dfferent varances, e can get a good approxmaton n 58 teratons (see Fgure.4). The dar sde of the EM algorthm s that t s very slo and s consdered to be mpractcal for large data sets. Fgure.4. Threecomponent mxture th to components havng equal means. The EM algorthm as nvented n late 970s, but the 990s mared a ne ave of research revved by grong nterest to data mnng. Recent research s gong n to drectons: () Generalzaton of the EM algorthm and extenson to other probablty dstrbutons, and () Improvng the speed of EM algorthm usng herarchcal data representatons..3 References and Applcatons There s a suffcent body of lterature devoted to the mxture models. Most of t s ntended for researches th a strong mathematcal bacground. I refer to to boos [34] rtten by Australan researchers. There are also several commercal and free softare pacages devoted to mxture models, most of hch are avalable on the Web. Commercal softare: MPUS MIX
10 Freeare: EMMIX MCUST MIXCUS NORMIX CAMAN Mxture Models have been n use for more than 30 years. It s a very poerful approach to modelbased clusterng and classfcaton. It has been appled to numerous problems. Some of the more nterestng applcatons are descrbed belo. Mxture models are dely used n mage processng for mage representaton and segmentaton, obect recognton, and contentbased mage retreval : Mxture models also ere used for text classfcaton and clusterng: Gaussan mxture models proved to be very successful for solvng the speaer recognton and speaer verfcaton problems: There s grong nterest n applyng ths technque to exploratory analyss of data or data mnng, and medcal dagnoss: You can learn more about these applcatons by follong the URs presented above. References [] J.R. Norrs (997) Marov Chans. Cambrdge Unversty Press. [] W.J. Steart (994) Introducton to the Numercal Solutons of Marov Chans. Prnceton Unversty Press. [3] G.J. Mcachlan and.e. Basford (988) Mxture Models: Inference and Applcaton to Clusterng. Ne Yor: Marcel Deer. [4] G.J. Mcachlan and T. rshnan (997) The EM Algorthm and Extensons. Ne Yor: Wley.
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